Bending of plates

Short description: Deformation of slabs under load

Bending of an edge-clamped circular plate under the action of a transverse pressure. The left half of the plate shows the deformed shape, while the right half shows the undeformed shape. This calculation was performed using Ansys.

Bending of plates, or plate bending, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the plate can be calculated from these deflections. Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load.

## Contents

* 1 Bending of Kirchhoff-Love plates
* 1.1 Definitions
* 1.1.1 Moments
* 1.1.2 Forces
* 1.1.3 Stresses
* 1.1.4 Strains
* 1.1.5 Deflections
* 1.2 Derivation
* 1.3 Small deflection of thin rectangular plates
* 1.4 Large deflection of thin rectangular plates
* 2 Circular Kirchhoff-Love plates
* 2.1 Clamped edges
* 3 Rectangular Kirchhoff-Love plates
* 3.1 Sinusoidal load
* 3.2 Navier solution
* 3.2.1 Double trigonometric series equation
* 3.2.2 Simply-supported plate with general load
* 3.2.3 Simply-supported plate with uniformly-distributed load
* 3.3 Lévy solution
* 3.3.1 Moments along edges
* 3.3.2 Simply-supported plate with uniformly-distributed load
* 3.3.3 Uniform and symmetric moment load
* 3.4 Cylindrical plate bending
* 3.4.1 Simply supported plate with axially fixed ends
* 4 Bending of thick Mindlin plates
* 4.1 Governing equations
* 4.2 Simply supported rectangular plates
* 5 Bending of Reissner-Stein cantilever plates
* 6 See also
* 7 References

## Bending of Kirchhoff-Love plates

Forces and moments on a flat plate.

### Definitions

For a thin rectangular plate of thickness H, Young's modulus E, and Poisson's ratio ν, we can define parameters in terms of the plate deflection, w.

The flexural rigidity is given by

D=EH312(1−ν2)

#### Moments

The bending moments per unit length are given by

Mx=−D(∂2w∂x2+ν∂2w∂y2)
My=−D(ν∂2w∂x2+∂2w∂y2)

The twisting moment per unit length is given by

Mxy=−D(1−ν)∂2w∂x∂y

#### Forces

The shear forces per unit length are given by

Qx=−D∂∂x(∂2w∂x2+∂2w∂y2)
Qy=−D∂∂y(∂2w∂x2+∂2w∂y2)

#### Stresses

The bending stresses are given by

σx=−12DzH3(∂2w∂x2+ν∂2w∂y2)
σy=−12DzH3(ν∂2w∂x2+∂2w∂y2)

The shear stress is given by

τxy=−12DzH3(1−ν)∂2w∂x∂y

#### Strains

The bending strains for small-deflection theory are given by

ϵx=∂u∂x=−z∂2w∂x2
ϵy=∂v∂y=−z∂2w∂y2

The shear strain for small-deflection theory is given by

γxy=∂u∂y+∂v∂x=−2z∂2w∂x∂y

For large-deflection plate theory, we consider the inclusion of membrane strains

ϵx=∂u∂x+12(∂w∂x)2
ϵy=∂v∂y+12(∂w∂y)2
γxy=∂u∂y+∂v∂x+∂w∂x∂w∂y

#### Deflections

The deflections are given by

u=−z∂w∂x
v=−z∂w∂y

### Derivation

In the Kirchhoff–Love plate theory for plates the governing equations are[1]

Nαβ,α=0

and

Mαβ,αβ−q=0

In expanded form,

∂N11∂x1+∂N21∂x2=0;∂N12∂x1+∂N22∂x2=0

and

∂2M11∂x12+2∂2M12∂x1∂x2+∂2M22∂x22=q

where q(x) is an applied transverse load per unit area, the thickness of the plate is H=2h, the stresses are σij, and

Nαβ:=∫−hhσαβdx3;Mαβ:=∫−hhx3σαβdx3.

The quantity N has units of force per unit length. The quantity M has units of moment per unit length.

For isotropic, homogeneous, plates with Young's modulus E and Poisson's ratio ν these equations reduce to[2]

∇2∇2w=−qD;D:=2h3E3(1−ν2)=H3E12(1−ν2)

where w(x1,x2) is the deflection of the mid-surface of the plate.

### Small deflection of thin rectangular plates

This is governed by the Germain-Lagrange plate equation

∂4w∂x4+2∂4w∂x2∂y2+∂4w∂y4=qD

This equation was first derived by Lagrange in December 1811 in correcting the work of Germain who provided the basis of the theory.

### Large deflection of thin rectangular plates

This is governed by the Föppl–von Kármán plate equations

∂4F∂x4+2∂4F∂x2∂y2+∂4F∂y4=E[(∂2w∂x∂y)2−∂2w∂x2∂2w∂y2]
∂4w∂x4+2∂4w∂x2∂y2+∂4w∂y4=qD+HD(∂2F∂y2∂2w∂x2+∂2F∂x2∂2w∂y2−2∂2F∂x∂y∂2w∂x∂y)

where F is the stress function.

## Circular Kirchhoff-Love plates

The bending of circular plates can be examined by solving the governing equation with appropriate boundary conditions. These solutions were first found by Poisson in 1829. Cylindrical coordinates are convenient for such problems. Here z is the distance of a point from the midplane of the plate.

The governing equation in coordinate-free form is

∇2∇2w=−qD.

In cylindrical coordinates (r,θ,z),

∇2w≡1r∂∂r(r∂w∂r)+1r2∂2w∂θ2+∂2w∂z2.

For symmetrically loaded circular plates, w=w(r), and we have

∇2w≡1rddr(rdwdr).

Therefore, the governing equation is

1rddr[rddr{1rddr(rdwdr)}]=−qD.

If q and D are constant, direct integration of the governing equation gives us

>     w(r)=−qr464D+C1ln⁡r+C2r22+C3r24(2ln⁡r−1)+C4

where Ci are constants. The slope of the deflection surface is

ϕ(r)=dwdr=−qr316D+C1r+C2r+C3rln⁡r.

For a circular plate, the requirement that the deflection and the slope of the deflection are finite at r=0 implies that C1=0. However, C3 need not equal 0, as the limit of rln⁡r exists as you approach r=0 from the right. However, if C3≠0, the second derivative of w will diverge at the center, corresponding to a point-like defect.

### Clamped edges

For a circular plate with clamped edges, we have w(a)=0 and ϕ(a)=0 at the edge of the plate (radius a). Using these boundary conditions and setting C3=0, we get

>     w(r)=−q64D(a2−r2)2andϕ(r)=qr16D(a2−r2).

The in-plane displacements in the plate are

ur(r)=−zϕ(r)anduθ(r)=0.

The in-plane strains in the plate are

εrr=durdr=−qz16D(a2−3r2),εθθ=urr=−qz16D(a2−r2),εrθ=0.

The in-plane stresses in the plate are

σrr=E1−ν2[εrr+νεθθ];σθθ=E1−ν2[εθθ+νεrr];σrθ=0.

For a plate of thickness 2h, the bending stiffness is D=2Eh3/[3(1−ν2)] and we have

>     σrr=−3qz32h3[(1+ν)a2−(3+ν)r2]σθθ=−3qz32h3[(1+ν)a2−(1+3ν)r2]σrθ=0.

The moment resultants (bending moments) are

Mrr=−q16[(1+ν)a2−(3+ν)r2];Mθθ=−q16[(1+ν)a2−(1+3ν)r2];Mrθ=0.

The maximum radial stress is at z=h and r=a:

σrr|z=h,r=a=3qa216h2=3qa24H2

where H:=2h. The bending moments at the boundary and the center of the plate are

Mrr|r=a=qa28,Mθθ|r=a=νqa28,Mrr|r=0=Mθθ|r=0=−(1+ν)qa216.

## Rectangular Kirchhoff-Love plates

Bending of a rectangular plate under the action of a distributed force q per unit area.

For rectangular plates, Navier in 1820 introduced a simple method for finding the displacement and stress when a plate is simply supported. The idea was to express the applied load in terms of Fourier components, find the solution for a sinusoidal load (a single Fourier component), and then superimpose the Fourier components to get the solution for an arbitrary load.

### Sinusoidal load

Let us assume that the load is of the form

q(x,y)=q0sin⁡πxasin⁡πyb.

Here q0 is the amplitude, a is the width of the plate in the x-direction, and b is the width of the plate in the y-direction.

Since the plate is simply supported, the displacement w(x,y) along the edges of the plate is zero, the bending moment Mxx is zero at x=0 and x=a, and Myy is zero at y=0 and y=b.

If we apply these boundary conditions and solve the plate equation, we get the solution

w(x,y)=q0π4D(1a2+1b2)−2sin⁡πxasin⁡πyb.

Where D is the flexural rigidity

D=Et312(1−ν2)

Analogous to flexural stiffness EI.[3] We can calculate the stresses and strains in the plate once we know the displacement.

For a more general load of the form

q(x,y)=q0sin⁡mπxasin⁡nπyb

where m and n are integers, we get the solution

>     (1)w(x,y)=q0π4D(m2a2+n2b2)−2sin⁡mπxasin⁡nπyb.

### Navier solution

#### Double trigonometric series equation

We define a general load q(x,y) of the following form

q(x,y)=∑m=1∞∑n=1∞amnsin⁡mπxasin⁡nπyb

where amn is a Fourier coefficient given by

amn=4ab∫0b∫0aq(x,y)sin⁡mπxasin⁡nπybdxdy.

The classical rectangular plate equation for small deflections thus becomes:

∂4w∂x4+2∂4w∂x2∂y2+∂4w∂y4=1D∑m=1∞∑n=1∞amnsin⁡mπxasin⁡nπyb

#### Simply-supported plate with general load

We assume a solution w(x,y) of the following form

w(x,y)=∑m=1∞∑n=1∞wmnsin⁡mπxasin⁡nπyb

The partial differentials of this function are given by

∂4w∂x4=∑m=1∞∑n=1∞(mπa)4wmnsin⁡mπxasin⁡nπyb
∂4w∂x2∂y2=∑m=1∞∑n=1∞(mπa)2(nπb)2wmnsin⁡mπxasin⁡nπyb
∂4w∂y4=∑m=1∞∑n=1∞(nπb)4wmnsin⁡mπxasin⁡nπyb

Substituting these expressions in the plate equation, we have

∑m=1∞∑n=1∞((mπa)2+(nπb)2)2wmnsin⁡mπxasin⁡nπyb=∑m=1∞∑n=1∞amnDsin⁡mπxasin⁡nπyb

Equating the two expressions, we have

((mπa)2+(nπb)2)2wmn=amnD

which can be rearranged to give

wmn=1π4Damn(m2a2+n2b2)2

The deflection of a simply-supported plate (of corner-origin) with general load is given by

>     w(x,y)=1π4D∑m=1∞∑n=1∞amn(m2a2+n2b2)2sin⁡mπxasin⁡nπyb

#### Simply-supported plate with uniformly-distributed load

Displacement (w)

Stress (σxx)

Stress (σyy)

Displacement and stresses along x=a/2 for a rectangular plate with a=20 mm, b=40 mm, H=2h=0.4 mm, E=70 GPa, and ν=0.35 under a load q0=−10 kPa. The red line represents the bottom of the plate, the green line the middle, and the blue line the top of the plate.

For a uniformly-distributed load, we have

q(x,y)=q0

The corresponding Fourier coefficient is thus given by

amn=4ab∫0a∫0bq0sin⁡mπxasin⁡nπybdxdy.

Evaluating the double integral, we have

amn=4q0π2mn(1−cos⁡mπ)(1−cos⁡nπ),

or alternatively in a piecewise format, we have

amn={16q0π2mnmandnodd0morneven

The deflection of a simply-supported plate (of corner-origin) with uniformly-distributed load is given by

>     w(x,y)=16q0π6D∑m=1,3,5,...∞∑n=1,3,5,...∞1mn(m2a2+n2b2)2sin⁡mπxasin⁡nπyb

The bending moments per unit length in the plate are given by

>     Mx=16q0π4∑m=1,3,5,...∞∑n=1,3,5,...∞m2a2+νn2b2mn(m2a2+n2b2)2sin⁡mπxasin⁡nπyb
>     My=16q0π4∑m=1,3,5,...∞∑n=1,3,5,...∞n2b2+νm2a2mn(m2a2+n2b2)2sin⁡mπxasin⁡nπyb

### Lévy solution

Another approach was proposed by Lévy[4] in 1899. In this case we start with an assumed form of the displacement and try to fit the parameters so that the governing equation and the boundary conditions are satisfied. The goal is to find Ym(y) such that it satisfies the boundary conditions at y=0 and y=b and, of course, the governing equation ∇2∇2w=q/D.

Let us assume that

w(x,y)=∑m=1∞Ym(y)sin⁡mπxa.

For a plate that is simply-supported along x=0 and x=a, the boundary conditions are w=0 and Mxx=0. Note that there is no variation in displacement along these edges meaning that ∂w/∂y=0 and ∂2w/∂y2=0, thus reducing the moment boundary condition to an equivalent expression ∂2w/∂x2=0.

#### Moments along edges

Consider the case of pure moment loading. In that case q=0 and w(x,y) has to satisfy ∇2∇2w=0. Since we are working in rectangular Cartesian coordinates, the governing equation can be expanded as

∂4w∂x4+2∂4w∂x2∂y2+∂4w∂y4=0.

Plugging the expression for w(x,y) in the governing equation gives us

∑m=1∞[(mπa)4Ymsin⁡mπxa−2(mπa)2d2Ymdy2sin⁡mπxa+d4Ymdy4sin⁡mπxa]=0

d4Ymdy4−2m2π2a2d2Ymdy2+m4π4a4Ym=0.

This is an ordinary differential equation which has the general solution

Ym=Amcosh⁡mπya+Bmmπyacosh⁡mπya+Cmsinh⁡mπya+Dmmπyasinh⁡mπya

where Am,Bm,Cm,Dm are constants that can be determined from the boundary conditions. Therefore, the displacement solution has the form

>     w(x,y)=∑m=1∞[(Am+Bmmπya)cosh⁡mπya+(Cm+Dmmπya)sinh⁡mπya]sin⁡mπxa.

Let us choose the coordinate system such that the boundaries of the plate are at x=0 and x=a (same as before) and at y=±b/2 (and not y=0 and y=b). Then the moment boundary conditions at the y=±b/2 boundaries are

w=0,−D∂2w∂y2|y=b/2=f1(x),−D∂2w∂y2|y=−b/2=f2(x)

where f1(x),f2(x) are known functions. The solution can be found by applying these boundary conditions. We can show that for the symmetrical case where

Myy|y=−b/2=Myy|y=b/2

and

f1(x)=f2(x)=∑m=1∞Emsin⁡mπxa

we have

>     w(x,y)=a22π2D∑m=1∞Emm2cosh⁡αmsin⁡mπxa(αmtanh⁡αmcosh⁡mπya−mπyasinh⁡mπya)

where

αm=mπb2a.

Similarly, for the antisymmetrical case where

Myy|y=−b/2=−Myy|y=b/2

we have

>     w(x,y)=a22π2D∑m=1∞Emm2sinh⁡αmsin⁡mπxa(αmcoth⁡αmsinh⁡mπya−mπyacosh⁡mπya).

We can superpose the symmetric and antisymmetric solutions to get more general solutions.

#### Simply-supported plate with uniformly-distributed load

For a uniformly-distributed load, we have

q(x,y)=q0

The deflection of a simply-supported plate with centre (a2,0) with uniformly-distributed load is given by

>     w(x,y)=q0a4D∑m=1,3,5,...∞(Amcosh⁡mπya+Bmmπyasinh⁡mπya+Gm)sin⁡mπxawhereAm=−2(αmtanh⁡αm+2)π5m5cosh⁡αmBm=2π5m5cosh⁡αmGm=4π5m5andαm=mπb2a

The bending moments per unit length in the plate are given by

>     Mx=−q0π2a2∑m=1,3,5,...∞m2(((ν−1)Am+2νBm)cosh⁡mπya+(ν−1)Bmmπyasinh⁡mπya−Gm)sin⁡mπxa
>     My=−q0π2a2∑m=1,3,5,...∞m2(((1−ν)Am+2Bm)cosh⁡mπya+(1−ν)Bmmπyasinh⁡mπya−νGm)sin⁡mπxa

#### Uniform and symmetric moment load

For the special case where the loading is symmetric and the moment is uniform, we have at y=±b/2,

Myy=f1(x)=4M0π∑m=1∞12m−1sin⁡(2m−1)πxa.

Displacement (w)

Bending stress (σyy)

Transverse shear stress (σyz)

Displacement and stresses for a rectangular plate under uniform bending moment along the edges y=−b/2 and y=b/2. The bending stress σyy is along the bottom surface of the plate. The transverse shear stress σyz is along the mid-surface of the plate.

The resulting displacement is

>     w(x,y)=2M0a2π3D∑m=1∞1(2m−1)3cosh⁡αmsin⁡(2m−1)πxa×[αmtanh⁡αmcosh⁡(2m−1)πya−(2m−1)πyasinh⁡(2m−1)πya]

where

αm=π(2m−1)b2a.

The bending moments and shear forces corresponding to the displacement w are

Mxx=−D(∂2w∂x2+ν∂2w∂y2)=2M0(1−ν)π∑m=1∞1(2m−1)cosh⁡αm×sin⁡(2m−1)πxa×[−(2m−1)πyasinh⁡(2m−1)πya+{2ν1−ν+αmtanh⁡αm}cosh⁡(2m−1)πya]Mxy=(1−ν)D∂2w∂x∂y=−2M0(1−ν)π∑m=1∞1(2m−1)cosh⁡αm×cos⁡(2m−1)πxa×[(2m−1)πyacosh⁡(2m−1)πya+(1−αmtanh⁡αm)sinh⁡(2m−1)πya]Qzx=∂Mxx∂x−∂Mxy∂y=4M0a∑m=1∞1cosh⁡αm×cos⁡(2m−1)πxacosh⁡(2m−1)πya.

The stresses are

σxx=12zh3Mxxandσzx=1κhQzx(1−4z2h2).

### Cylindrical plate bending

Cylindrical bending occurs when a rectangular plate that has dimensions a×b×h, where a≪b and the thickness h is small, is subjected to a uniform distributed load perpendicular to the plane of the plate. Such a plate takes the shape of the surface of a cylinder.

#### Simply supported plate with axially fixed ends

For a simply supported plate under cylindrical bending with edges that are free to rotate but have a fixed x1. Cylindrical bending solutions can be found using the Navier and Levy techniques.

## Bending of thick Mindlin plates

For thick plates, we have to consider the effect of through-the-thickness shears on the orientation of the normal to the mid-surface after deformation. Raymond D. Mindlin's theory provides one approach for find the deformation and stresses in such plates. Solutions to Mindlin's theory can be derived from the equivalent Kirchhoff-Love solutions using canonical relations.[5]

### Governing equations

The canonical governing equation for isotropic thick plates can be expressed as[5]

∇2(ℳ−ℬ1+νq)=−qκGh(∇2w+ℳD)=−(1−ℬc21+ν)q∇2(∂φ1∂x2−∂φ2∂x1)=c2(∂φ1∂x2−∂φ2∂x1)

where q is the applied transverse load, G is the shear modulus, D=Eh3/[12(1−ν2)] is the bending rigidity, h is the plate thickness, c2=2κGh/[D(1−ν)], κ is the shear correction factor, E is the Young's modulus, ν is the Poisson's ratio, and

ℳ=D[𝒜(∂φ1∂x1+∂φ2∂x2)−(1−𝒜)∇2w]+2q1−ν2ℬ.

In Mindlin's theory, w is the transverse displacement of the mid-surface of the plate and the quantities φ1 and φ2 are the rotations of the mid-surface normal about the x2 and x1-axes, respectively. The canonical parameters for this theory are 𝒜=1 and ℬ=0. The shear correction factor κ usually has the value 5/6.

The solutions to the governing equations can be found if one knows the corresponding Kirchhoff-Love solutions by using the relations

w=wK+ℳKκGh(1−ℬc22)−Φ+Ψφ1=−∂wK∂x1−1κGh(1−1𝒜−ℬc22)Q1K+∂∂x1(DκGh𝒜∇2Φ+Φ−Ψ)+1c2∂Ω∂x2φ2=−∂wK∂x2−1κGh(1−1𝒜−ℬc22)Q2K+∂∂x2(DκGh𝒜∇2Φ+Φ−Ψ)+1c2∂Ω∂x1

where wK is the displacement predicted for a Kirchhoff-Love plate, Φ is a biharmonic function such that ∇2∇2Φ=0, Ψ is a function that satisfies the Laplace equation, ∇2Ψ=0, and

ℳ=ℳK+ℬ1+νq+D∇2Φ;ℳK:=−D∇2wKQ1K=−D∂∂x1(∇2wK),Q2K=−D∂∂x2(∇2wK)Ω=∂φ1∂x2−∂φ2∂x1,∇2Ω=c2Ω.

### Simply supported rectangular plates

For simply supported plates, the Marcus moment sum vanishes, i.e.,

ℳ=11+ν(M11+M22)=D(∂φ1∂x1+∂φ2∂x2)=0.

Which is almost Laplace`s equation for w[ref 6]. In that case the functions Φ, Ψ, Ω vanish, and the Mindlin solution is related to the corresponding Kirchhoff solution by

w=wK+ℳKκGh.

## Bending of Reissner-Stein cantilever plates

Reissner-Stein theory for cantilever plates[6] leads to the following coupled ordinary differential equations for a cantilever plate with concentrated end load qx(y) at x=a.

bDd4wxdx4=0b3D12d4θxdx4−2bD(1−ν)d2θxdx2=0

and the boundary conditions at x=a are

bDd3wxdx3+qx1=0,b3D12d3θxdx3−2bD(1−ν)dθxdx+qx2=0bDd2wxdx2=0,b3D12d2θxdx2=0.

Solution of this system of two ODEs gives

wx(x)=qx16bD(3ax2−x3)θx(x)=qx22bD(1−ν)[x−1νb(sinh⁡(νba)cosh⁡[νb(x−a)]+tanh⁡[νb(x−a)])]

where νb=24(1−ν)/b. The bending moments and shear forces corresponding to the displacement w=wx+yθx are

Mxx=−D(∂2w∂x2+ν∂2w∂y2)=qx1(x−ab)−[3yqx2b3νbcosh3[νb(x−a)]]×[6sinh⁡(νba)−sinh⁡[νb(2x−a)]+sinh⁡[νb(2x−3a)]+8sinh⁡[νb(x−a)]]Mxy=(1−ν)D∂2w∂x∂y=qx22b[1−2+cosh⁡[νb(x−2a)]−cosh⁡[νbx]2cosh2[νb(x−a)]]Qzx=∂Mxx∂x−∂Mxy∂y=qx1b−(3yqx22b3cosh4[νb(x−a)])×[32+cosh⁡[νb(3x−2a)]−cosh⁡[νb(3x−4a)]−16cosh⁡[2νb(x−a)]+23cosh⁡[νb(x−2a)]−23cosh⁡(νbx)].

The stresses are

σxx=12zh3Mxxandσzx=1κhQzx(1−4z2h2).

If the applied load at the edge is constant, we recover the solutions for a beam under a concentrated end load. If the applied load is a linear function of y, then

qx1=∫−b/2b/2q0(12−yb)dy=bq02;qx2=∫−b/2b/2yq0(12−yb)dy=−b2q012.

## See also

* Bending
* Infinitesimal strain theory
* Kirchhoff–Love plate theory
* Linear elasticity
* Mindlin–Reissner plate theory
* Plate theory
* Stress (mechanics)
* Stress resultants
* Structural acoustics
* Vibration of plates

## References

1. ↑ Reddy, J. N., 2007, Theory and analysis of elastic plates and shells, CRC Press, Taylor and Francis.
2. ↑ Timoshenko, S. and Woinowsky-Krieger, S., (1959), Theory of plates and shells, McGraw-Hill New York.
3. ↑ Cook, R. D. et al., 2002, Concepts and applications of finite element analysis, John Wiley & Sons
4. ↑ Lévy, M., 1899, Comptes rendues, vol. 129, pp. 535-539
5. ↑ 5.0 5.1 Lim, G. T. and Reddy, J. N., 2003, On canonical bending relationships for plates, International Journal of Solids and Structures, vol. 40, pp. 3039-3067.
6. ↑ E. Reissner and M. Stein. Torsion and transverse bending of cantilever plates. Technical Note 2369, National Advisory Committee for Aeronautics,Washington, 1951.

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